A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In that instance, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g. including part of, one, or several dies).on a substrate (e.g. a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at once, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti-parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate.
An important factor in lithographic apparatus performance is the precision with which components to be moved during exposure, such as the reticle stage (patterning device table) containing the patterns needed for illumination and the substrate table containing the substrates to be illuminated, can be displaced. Under feedback control, the movement of components is controlled using standard PID-based control systems. However, to obtain nano-scale position accuracy, with settling times of the order of milliseconds or lower, feedforward control may be desirable.
In addition to the commonly used acceleration-, jerk-, and even snap-based feedforward control designs (i.e. designs based on acceleration and higher order derivatives of position with respect to time), the application of iterative learning control to obtain short settling times has been suggested. This approach has the benefit that only limited system knowledge is required to implement the feedforward control with high accuracy. The method is based on iteratively learning a feedforward signal or “force” that minimizes a measured error signal (defined as a measured deviation of the state of a component being moved from a setpoint profile defining an intended time evolution of the state) over a number of trial “runs” of the component through the setpoint profile. When the learned signal is applied to the system or process, it effectively counteracts contributions to the error signal that occur repeatedly in different trials (“repetitive contributions”).
During learning of the feedforward signal, the measured error signal during a particular trial may contain non-repetitive contributions, like random noise, which differ from trial to trial. Such contributions may cause the learned feedforward signal to inject noise into the system. This may lead to a decrease in performance and/or limit the improvement obtained using iterative learning-based control. The efficiency of the learning process itself depends on the gain of the learning algorithm, which may be limited by its stability.